課程資訊
課程名稱
微分幾何二
Differential Geometry (Ⅱ) 
開課學期
106-2 
授課對象
理學院  數學研究所  
授課教師
蔡宜洵 
課號
MATH7302 
課程識別碼
221 U2940 
班次
 
學分
3.0 
全/半年
半年 
必/選修
必修 
上課時間
星期三9(16:30~17:20)星期五3,4(10:20~12:10) 
上課地點
天數305天數305 
備註
研究所數學組基礎課。
總人數上限:40人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1062MATH7302_ 
課程簡介影片
 
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課程概述

1. Jacobi fields, 2nd variation, Jacobi equation, conjugate points, minimizing property of geodesics, *Index Lemma, *Jacobi’s theorem, two proofs
2. Myers-Bonnet theorem, Cartan-Hadamard theorem, *Rauch comparison theorem with applications to injectivity radius
3. Space of constant curvature, group theory viewpoints, geodesics, Jacobi fields
4. Cartan-Ambrose-Hicks Theorem
5, 6, 7. Miscellaneous*
a) flows and transformations
b) Killing vector fields
c) volume element and divergence
d) Ricci curvature and volume growth
e) 2nd Bianchi identity applied to Einstein manifolds
f) Cut locus, injectivity radius, Klingenberg’s lemma

8. vector bundles, bundle maps, pull-back bundles, complex vector bundles
9. connection, curvature form, Bianchi identity
10. Chern classes, invariant polynomials, Chern character, unitary connection
11. Examples and application of Chern classes, *immersions and embeddings in complex projective spaces
12. Pontrjagin classes, Euler class, relation with Chern classes,Todd class, A-hat genus
13. star operator, Hodge decomposition theorem, Poincare duality,
14. Kunneth formula, Bochner-Weitzenbock formula, proof
15. divergence, application of B-W formula to topology of manifolds, index of de Rham complex, remark on Index theorem
16. Gauge theory, Erlanger Program, historical remarks, principle bundles
*17. Examples of Lie groups, SU(2)-bundles, Yang-Mills equation, self-duality equation

Note. Items with * may be partly or completely skipped due to time constraint.
 

課程目標
Understand the fundamental tools and theorems in differential geometry, especially in Riemann geometry. 
課程要求
1.Homework 30%
2.Midterm exam 35%
3.Final exam 35% 
預期每週課後學習時數
 
Office Hours
 
參考書目
待補 
指定閱讀

References (those given in 1st semester and the following)

1. Taubes, C., “Differential geometry”: discusses bundle theory in great detail
2. Kobayashi-Nomizu: “Foundations of Differential geometry”: bundle, principle bundles and related topics.
3. Berline, N., Getzler, E., Vergne, M., “Heat kernels and Dirac operators”: a more specialized book on bundle theory and used it for various index theorems

 
評量方式
(僅供參考)
   
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